With every reduced E-Fountain semigroup S which satisfies the generalized right ample condition we associate a category with zero morphisms <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {C}(S)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Under some assumptions we prove an isomorphism of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Bbbk $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation>-algebras <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Bbbk S\simeq \Bbbk _{0}\mathcal {C}(S)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">k</mi> <mi>S</mi> <mo>≃</mo> <msub> <mi mathvariant="double-struck">k</mi> <mn>0</mn> </msub> <mi mathvariant="script">C</mi> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> between the semigroup algebra and the contracted category algebra where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Bbbk $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation> is any commutative unital ring. This is a simultaneous generalization of a former result of the author on reduced E-Fountain semigroups which satisfy the congruence condition, a result of Junying Guo and Xiaojiang Guo on strict right ample semigroups and a result of Benjamin Steinberg on idempotent semigroups with central idempotents. The applicability of the new isomorphism is demonstrated with two well-known monoids which are not members of the above classes. The monoid of order-preserving functions on an n-set and the monoid of binary relations with demonic composition.

The algebra of the monoid of order-preserving functions on an n-set and other reduced E-Fountain semigroups

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2025年11月20日

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Itamar Stein

摘要

With every reduced E-Fountain semigroup S which satisfies the generalized right ample condition we associate a category with zero morphisms $\mathcal {C}(S)$ $C (S)$ . Under some assumptions we prove an isomorphism of $\Bbbk $ $k$ -algebras $\Bbbk S\simeq \Bbbk _{0}\mathcal {C}(S)$ $k S ≃ k_{0} C (S)$ between the semigroup algebra and the contracted category algebra where $\Bbbk $ $k$ is any commutative unital ring. This is a simultaneous generalization of a former result of the author on reduced E-Fountain semigroups which satisfy the congruence condition, a result of Junying Guo and Xiaojiang Guo on strict right ample semigroups and a result of Benjamin Steinberg on idempotent semigroups with central idempotents. The applicability of the new isomorphism is demonstrated with two well-known monoids which are not members of the above classes. The monoid of order-preserving functions on an n-set and the monoid of binary relations with demonic composition.